Duality theory for nonergodic actions

TL;DR

This paper develops a duality framework for actions of compact quantum groups on C*-algebras using weak unitary tensor functors, connecting to crossed product constructions and module categories, with applications to algebra deformations.

Contribution

It introduces a novel characterization of quantum group actions via weak unitary tensor functors and links these to known crossed product constructions and module categories.

Findings

Characterization of quantum group actions using weak unitary tensor functors

Connection of the construction to crossed product and Fell bundle frameworks

Application to deformations of C*-algebras by quantum group cocycles

Abstract

Generalizing work by Pinzari and Roberts, we characterize actions of a compact quantum group G on C*-algebras in terms of what we call weak unitary tensor functors from Rep G into categories of C*-correspondences. We discuss the relation of our construction of a C*-algebra from a functor to some well-known crossed product type constructions, such as cross-sectional algebras of Fell bundles and crossed products by Hilbert bimodules. We also relate our setting to recent work of De Commer and Yamashita by showing that any object in a module C*-category over Rep G produces a weak unitary tensor functor, and, as a consequence, actions can also be described in terms of (Rep G)-module C*-categories. As an application we discuss deformations of C*-algebras by cocycles on discrete quantum groups.